Presenting higher stacks as simplicial schemes

Abstract

We show that an n-geometric stack may be regarded as a special kind of simplicial scheme, namely a Duskin n-hypergroupoid in affine schemes, where surjectivity is defined in terms of covering maps, yielding Artin n-stacks, Deligne-Mumford n-stacks and n-schemes as the notion of covering varies. This formulation adapts to all HAG contexts, so in particular works for derived n-stacks (replacing rings with simplicial rings). We exploit this to describe quasi-coherent sheaves and complexes on these stacks, and to draw comparisons with Kontsevich's dg-schemes. As an application, we show how the cotangent complex controls infinitesimal deformations of higher and derived stacks.

55 pages; v3 content rearranged with many corrections; final version, to appear in Adv. Math; v4 corrections in section 7.1

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• Notes characterising higher and derived stacks concretely
• The homology of moduli stacks of complexes
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• Augmented Homotopical Algebraic Geometry
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• Extended Field Theories as higher Kaluza-Klein theories
• A differential graded model for derived analytic geometry